Skip to main content
SpringerLink
Log in

On the Rate of Convergence in the Invariance Principle for Weakly Dependent Random Variables

  • Published:
Ukrainian Mathematical Journal Aims and scope

We consider nonstationary sequences of φ -mixing random variables. By using the Levy–Prokhorov distance, we estimate the rate of convergence in the invariance principle for nonstationary φ -mixing random variables. The obtained results extend and generalize several known facts established for nonstationary φ -mixing random variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. T. V. Arak, “An estimate of A. A. Borovkov,” Theory Probab. Appl., 20, No. 2, 380–381 (1976).

    Article  MathSciNet  Google Scholar 

  2. I. Berkes and W. Philipp, “Approximation theorems for independent and weakly dependent random vectors,” Ann. Probab., 7, No. 1, 29–54 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).

    MATH  Google Scholar 

  4. A. A. Borovkov, “On the convergence rate in the invariance principle,” Theory Probab. Appl., 29, No. 3, 550–553 (1985).

    Article  MATH  Google Scholar 

  5. A. A. Borovkov and A. I. Sakhanenko, “On the estimates of the rate of convergence in the invariance principle for Banach spaces,” Theory Probab. Appl., 25, No. 4, 734–744 (1981).

    Article  MATH  Google Scholar 

  6. R. C. Bradley, “Basic properties of strong mixing conditions. A survey and some open questions,” Probab. Surv., 2, 107–144 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  7. R. M. Dudley, “Distance of probability measures and random variables,” Ann. Math. Statist., 39, 1563–1572 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Donsker, “An invariance principle for certain probability limit theorems,” Mem. Amer. Math. Soc., 6, 250–268 (1951).

    MathSciNet  MATH  Google Scholar 

  9. V. V. Gorodetsky, “On the rate of convergence in the invariance principle for strongly mixing sequences,” Theory Probab. Appl., 28, No. 4, 780–785 (1983).

    MathSciNet  Google Scholar 

  10. C. C. Heyde, “Some properties of metrics on a study on convergence to normality,” Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 11, No. 3, 181–192 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  11. I. A. Ibragimov and V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen (1971).

    MATH  Google Scholar 

  12. S. Kanagawa, “Rates of convergence of the invariance principle for weakly dependent random variables,” Yokohama Math. J., 30, No. 1-2, 103–119 (1982).

    MathSciNet  MATH  Google Scholar 

  13. J. Komlos, P. Major, and G. Z. Tusnady, “An approximation of partial sums of independent RV’s and the sample DF. I,” Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 32, No. 2, 111–131 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Komlos, P. Major, and G. Z. Tusnady, “An approximation of partial sums of independent RV’s and the sample DF. II,” Z.Wahrscheinlichkeitstheorie Verw. Gebiete, 34, No. 1, 33–58 (1976).

  15. M. Peligrad and S. Utev, “A new maximal inequality and invariance principle for stationary sequences,” Ann. Probab., 33, No. 2, 798–815 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  16. Yu. V. Prokhorov, “Convergence of random processes and limit theorems in probability theory,” Theory Probab. Appl., 1, No. 2, 157–214 (1956).

    Article  MathSciNet  Google Scholar 

  17. A. I. Sakhanenko, “Estimates for the rate of convergence in the invariance principle,” Dokl. Akad. Nauk SSSR, 219, 1076–1078 (1974).

    MathSciNet  Google Scholar 

  18. A. I. Sakhanenko, “Estimates in the invariance principle,” Trudy Inst. Mat. SO RAN [in Russian], Vol. 5, Nauka, Novosibirsk (1985), pp. 27–44.

  19. A. I. Sakhanenko, “On the accuracy of normal approximation in the invariance principle,” Trudy Inst. Mat. SO RAN [in Russian], Vol. 19, Nauka, Novosibirsk (1989), pp. 40–66.

  20. A. I. Sakhanenko, “Estimates in the invariance principle in terms of truncated power moments,” Sib. Math. J., 47, No. 6, 1113–1127 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. I. Sakhanenko, “A general estimate in the invariance principle,” Sib. Math. J., 52, No. 4, 696–710 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. V. Skorokhod, Research on the Theory of Stochastic Processes, Kiev University Press, Kiev (1961).

    Google Scholar 

  23. S. A. Utev, “Inequalities for sums of weakly dependent random variables and estimates of rate of convergence in the invariance principle,” Limit Theorems for Sums of Random Variables, Tr. Inst. Mat. [in Russian], 3 (1984), pp. 50–77.

  24. S. A. Utev, “Sums of φ -mixing random variables,” Asymptotic Analysis of Distributions of Random Processes [in Russian], Nauka, Novosibirsk (1989), pp. 78–100.

    Google Scholar 

  25. K. Yoshihara, “Convergence rates of the invariance principle for absolutely regular sequence,” Yokohama Math. J., 27, No. 1, 49–55 (1979).

    MathSciNet  MATH  Google Scholar 

  26. T. M. Zuparov and A. K. Muhamedov, “An invariance principle for processes with uniformly strongly mixing,” Proc. Funct. Random Processes and Statistical Inference, Fan, Tashkent (1989), pp. 27–36.

  27. T. M. Zuparov and A. K. Muhamedov, “On the rate of convergence of the invariance principle for φ -mixing processes,” Proc. Rep. VI USSR–Japan Symp. Probab. Theory and Math. Statistics, Kiev, August 5–10 (1991), p. 65.

Download references

Author information

Authors and Affiliations

  1. National University of Uzbekistan, Tashkent, Uzbekistan

    A. K. Mukhamedov

Authors
  1. A. K. Mukhamedov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. K. Mukhamedov.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 9, pp. 1216–1230, September, 2022. Ukrainian DOI:https://doi.org/10.37863/umzh.v74i9.6244

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukhamedov, A.K. On the Rate of Convergence in the Invariance Principle for Weakly Dependent Random Variables. Ukr Math J 74, 1386–1403 (2023). https://doi.org/10.1007/s11253-023-02142-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-023-02142-5

Search

Navigation